'When Are Relations Neither True Nor False?
in [Math]
|
From: Frederick Williams on 23 Feb 2010 10:53 Newberry wrote: > > Frederick Williams wrote: > > > This is one of those threads that causes me to think "would that the > > > contributors could find something more interesting to discuss." > > > [...] Goedel's incompeteness theorem suggests that two valued > logic is impossible. Wow! If that were so I'd withdraw my remark.
From: Aatu Koskensilta on 23 Feb 2010 11:10 Newberry <newberryxy(a)gmail.com> writes: > Goedel's incompeteness theorem suggests that two valued logic is > impossible. Anything may suggest anything to someone. If we are to discuss this you need to spell out in some detail this suggestion, and what you mean by saying that "two valued logic is impossible". -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Nam Nguyen on 23 Feb 2010 21:38 Aatu Koskensilta wrote: > Newberry <newberryxy(a)gmail.com> writes: > >> Goedel's incompeteness theorem suggests that two valued logic is >> impossible. > > Anything may suggest anything to someone. If we are to discuss this you > need to spell out in some detail this suggestion, and what you mean by > saying that "two valued logic is impossible". > He (Newberry) might have a different idea, but how about "it's impossible to completely define some model relations". Would this be some good details you're looking for?
From: Nam Nguyen on 23 Feb 2010 22:52 Newberry wrote: > On Feb 22, 11:00 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Frederick Williams wrote: >>> This is one of those threads that causes me to think "would that the >>> contributors could find something more interesting to discuss." >> This thread to me is an interesting one in that (at least) its title >> suggests a way to generalize GIT into a more comprehensive statement >> about incompleteness in mathematical reasoning, through FOL. > > I do not know if I would call it generalization but it is something of > that sort. I'm not sure I get what you said here. If it's something of that sort of "generalization", why you wouldn't call it "generalization"? [But I don't think we have to argue about it so if you don't feel like answering the question that's fine with me.] > Goedel's incompeteness theorem suggests that two valued > logic is impossible. In what way though? > Furthermore Goedel's second theorem does not > apply to theories with gaps. This is the 2nd time (iirc) you mentioned "theories with gaps". What would you mean by "gaps"? > It is obvious where the gaps might be - > in the so called "vacuously true" sentences. > > So again, if > > ~(Ex)[(x + n < 6) & (n = 8)] > > is neither true nor false for any n (according to the logic of > presuppositions.) Am I right that > > ~(Ex)(Ey)[(x + y < 6) & (y = 8)] > > is neither true nor false? I actually don't know what "The logic of presuppositions" is so I couldn't comment on it. In FOL, depending on the exact axioms, you might have some consistent theories where the formula would be true in any model, and some others where it'd be be false. For example, in T with these axioms: A1: Ax[Sx=0] A2: Axy[x+y=0] A3: (x < y) <-> x=y then ~((Ex)(Ey)[(x + y < 6) & (y = 8)]) would be _false_ in any model of T.
From: Newberry on 24 Feb 2010 00:17
On Feb 23, 7:53 am, Frederick Williams <frederick.willia...(a)tesco.net> wrote: > Newberry wrote: > > > Frederick Williams wrote: > > > > This is one of those threads that causes me to think "would that the > > > > contributors could find something more interesting to discuss." > > > [...] Goedel's incompeteness theorem suggests that two valued > > logic is impossible. > > Wow! If that were so I'd withdraw my remark. What remark? |